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2 1. Fid the coordiates of the statioary poit o the curve with equatio y = 1. (4) Q1 (Total 4 marks) *N349B038* 3 Tur over

3 10. Figure 1 y 8 y = + 5 P R Q O Figure 1 shows part of the curve C with equatio 8 y = + 5, > 0. The poits P ad Q lie o C ad have -coordiates 1 ad 4 respectively. The regio R, shaded i Figure 1, is bouded by C ad the straight lie joiig P ad Q. (a) Fid the eact area of R. (b) Use calculus to show that y is icreasig for >. (8) (4) 4 *N349B048*

4 7. The curve C has equatio y = (a) Fid d y. d () (b) Usig the result from part (a), fid the coordiates of the turig poits of C. d y (c) Fid. d (d) Hece, or otherwise, determie the ature of the turig poits of C. (4) () () 14 *N355A0140*

5 Questio 7 cotiued Q7 (Total 10 marks) *N355A0150* 15 Tur over

6 10. Figure 3 y y = A R B O N Figure 3 shows a sketch of part of the curve with equatio y = The curve has statioary poits A ad B. (a) Use calculus to fid the -coordiates of A ad B. d y (b) Fid the value of at A, ad hece verify that A is a maimum. d (4) () The lie through B parallel to the y-ais meets the -ais at the poit N. The regio R, show shaded i Figure 3, is bouded by the curve, the -ais ad the lie from A to N. (c) Fid 3 ( 8 0 )d. (3) (d) Hece calculate the eact area of R. (5) 18 *N3558A0180*

7 1. f() = Fid (a) f ), (b) f ()d. 1 (3) (4) *N43A04*

8 8. A diesel lorry is drive from Birmigham to Bury at a steady speed of v kilometres per hour. The total cost of the jourey, C, is give by 1400 v C. v 7 (a) Fid the value of v for which C is a miimum. d C (b) Fid ad hece verify that C is a miimum for this value of v. dv (c) Calculate the miimum total cost of the jourey. (5) () () 16 *N43A0164*

9 physicsadmathstutor.com Jue cm y cm cm Figure 4 Figure 4 shows a solid brick i the shape of a cuboid measurig cm by cm by y cm. The total surface area of the brick is 600 cm. (a) Show that the volume, V cm 3, of the brick is give by 3 4 V = (4) Give that ca vary, (b) use calculus to fid the maimum value of V, givig your aswer to the earest cm 3. (5) (c) Justify that the value of V you have foud is a maimum. () *H6108A04*

10 physicsadmathstutor.com Jue 007 Questio 10 cotiued Q10 (Total 11 marks) TOTAL FOR PAPER: 75 MARKS END 4 *H6108A044*

11 9. Figure 4 y Figure 4 shows a ope-topped water tak, i the shape of a cuboid, which is made of sheet metal. The base of the tak is a rectagle metres by y metres. The height of the tak is metres. The capacity of the tak is 100 m 3. (a) Show that the area A m of the sheet metal used to make the tak is give by A = (b) Use calculus to fid the value of for which A is statioary. (c) Prove that this value of gives a miimum value of A. (d) Calculate the miimum area of sheet metal eeded to make the tak. (4) (4) () () *H630B04*

12 Questio 9 cotiued Q9 (Total 1 marks) TOTAL FOR PAPER: 75 MARKS END 4 *H630B044*

13 10. A solid right circular cylider has radius r cm ad height h cm. The total surface area of the cylider is 800 cm. (a) Show that the volume, V cm 3, of the cylider is give by V = 400r r 3. (4) Give that r varies, (b) use calculus to fid the maimum value of V, to the earest cm 3. (6) (c) Justify that the value of V you have foud is a maimum. () 6 *H30957A068*

14 Questio 10 cotiued Q10 (Total 1 marks) TOTAL FOR PAPER: 75 MARKS END 8 *H30957A088*

15 9. h 1 rad r r Figure Figure shows a closed bo used by a shop for packig pieces of cake. The bo is a right prism of height h cm. The cross sectio is a sector of a circle. The sector has radius r cm ad agle 1 radia. The volume of the bo is 300 cm 3. (a) Show that the surface area of the bo, cm, is give by 1800 r r (b) Use calculus to fid the value of r for which is statioary. (5) (4) (c) Prove that this value of r gives a miimum value of. () (d) Fid, to the earest cm, this miimum value of. () *H3463A04*

16 Questio 9 cotiued Q9 END (Total 13 marks) TOTAL FOR PAPER: 75 MARKS 4 *H3463A044*

17 3 9. The curve C has equatio y = 1 ( ) 10, 0 (a) Use calculus to fid the coordiates of the turig poit o C. (b) Fid d y d. (7) () (c) State the ature of the turig poit. (1) *N35101A04*

18 Questio 9 cotiued Q9 (Total 10 marks) TOTAL FOR PAPER: 75 MARKS END 4 *N35101A044*

19 3. y k, where is a costat. (a) Fid d y. d () (b) Give that y is decreasig at 4, fid the set of possible values of. () 6 *H35384A068*

20 10. The volume V cm 3 of a bo, of height cm, is give by V = 4 (5 ), 0 5 (a) Fid d V d. (b) Hece fid the maimum volume of the bo. (4) (4) (c) Use calculus to justify that the volume that you foud i part (b) is a maimum. () 6 *H35403A068*

21 Questio 10 cotiued Q10 END (Total 10 marks) TOTAL FOR PAPER: 75 MARKS 8 *H35403A088*

22 8. Figure A cuboid has a rectagular cross-sectio where the legth of the rectagle is equal to twice its width, cm, as show i Figure. The volume of the cuboid is 81 cubic cetimetres. (a) Show that the total legth, L cm, of the twelve edges of the cuboid is give by L = (b) Use calculus to fid the miimum value of L. (3) (6) (c) Justify, by further differetiatio, that the value of L that you have foud is a miimum. () 4 *P38158A043*

23 Questio 8 cotiued *P38158A053* 5 Tur over

24 physicsadmathstutor.com Jauary y Figure 3 Figure 3 shows a flowerbed. Its shape is a quarter of a circle of radius metres with two equal rectagles attached to it alog its radii. Each rectagle has legth equal to metres ad width equal to y metres. Give that the area of the flowerbed is 4 m, (a) show that 16 ϖ y = 8 (3) (b) Hece show that the perimeter P metres of the flowerbed is give by the equatio 8 P = + (c) Use calculus to fid the miimum value of P. (3) (5) (d) Fid the width of each rectagle whe the perimeter is a miimum. Give your aswer to the earest cetimetre. () *P40083A08*

25 physicsadmathstutor.com Jauary 01 Questio 8 cotiued *P40083A038* 3 Tur over

26 physicsadmathstutor.com Jue h mm Figure 3 mm A maufacturer produces pai relievig tablets. Each tablet is i the shape of a solid circular cylider with base radius mm ad height h mm, as show i Figure 3. Give that the volume of each tablet has to be 60 mm 3, (a) epress h i terms of, (b) show that the surface area, A mm, of a tablet is give by A = + 10 (1) (3) The maufacturer eeds to miimise the surface area A mm, of a tablet. (c) Use calculus to fid the value of for which A is a miimum. (d) Calculate the miimum value of A, givig your aswer to the earest iteger. (5) () (e) Show that this value of A is a miimum. () *P40685A08*

27 physicsadmathstutor.com Jue 01 Questio 8 cotiued *P40685A038* 3 Tur over

28 physicsadmathstutor.com Jauary The curve C has equatio y = 6 3 3, 0 (a) Use calculus to show that the curve has a turig poit P whe = (b) Fid the -coordiate of the other turig poit Q o the curve. (c) Fid d y d. (4) (1) (1) (d) Hece or otherwise, state with justificatio, the ature of each of these turig poits P ad Q. (3) 4 *P41487A043*

29 physicsadmathstutor.com Jauary 013 Questio 8 cotiued *P41487A053* 5 Tur over

30 physicsadmathstutor.com Jue 013 (R) 1. Usig calculus, fid the coordiates of the statioary poit o the curve with equatio y 8 = > 0, (6) *P486A03*

31 physicsadmathstutor.com Jue The curve with equatio has a statioary poit P. y = 3 ) Use calculus (a) to fid the coordiates of P, (6) (b) to determie the ature of the statioary poit P. (3) 6 *P41859A063*

32 Edecel AS/A level Mathematics Formulae List: Core Mathematics C Issue 1 September Core Mathematics C Cadidates sittig C may also require those formulae listed uder Core Mathematics C1. Cosie rule a = b + c bc cos A Biomial series 1 ) ( 1 r r b b a r b a b a a b a = + K K ( N) where )!!(! C r r r r = = < = + r r r 1, ( 1 1) ( 1) ( 1 1) ( 1 ) (1 K K K K R) Logarithms ad epoetials a b b a log log log = Geometric series u = ar 1 S = r r a 1 ) (1 S = r a 1 for r < 1 Numerical itegratio The trapezium rule: b a y d 1 h{(y 0 + y ) + (y 1 + y y 1 )}, where a b h =

33 Core Mathematics C1 Mesuratio Surface area of sphere = 4π r Area of curved surface of coe = π r slat height Arithmetic series u = a + ( 1)d S = 1 (a + l) = 1 [a + ( 1)d] 4 Edecel AS/A level Mathematics Formulae List: Core Mathematics C1 Issue 1 September 009

physicsandmathstutor.com

physicsandmathstutor.com physicsadmathstutor.com 8. The circle C, with cetre at the poit A, has equatio x 2 + y 2 10x + 9 = 0. Fid (a) the coordiates of A, (b) the radius of C, (c) the coordiates of the poits at which C crosses